1.06g Equations with exponentials: solve a^x = b

3 Solve the equation \(6 ^ { 2 x - 1 } = 3 ^ { x + 2 }\), giving your answer in the form \(x = \frac { \ln a } { \ln b }\) where \(a\) and \(b\) are integers.

6 Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time \(t\) hours after the injection, the concentration of Antiflu in Diane's bloodstream is \(3 \mathrm { e } ^ { - 0.02 t }\) units and the concentration of Coldcure is \(5 \mathrm { e } ^ { - 0.07 t }\) units. Each drug becomes ineffective when its concentration falls below 1 unit.

1. Show that Coldcure becomes ineffective before Antiflu.
2. Sketch, on the same diagram, the graphs of concentration against time for each drug.
3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later.

1 It is given that \(8 ^ { 4 x } = 4 ^ { 3 x - 6 }\).

1. By expressing each side as a power of 2 , find the value of \(x\).
2. Write down the value of \(\log _ { 4 } | x |\).

Use logarithms to solve the equation \(6^{2x-1} = 5e^{3x+2}\). Give your answer correct to 4 significant figures. [4]

Use logarithms to solve the equation \(3^{4t+3} = 5^{2t+7}\). Give your answer correct to 3 significant figures. [4]

The polynomial \(\mathrm{p}(x)\) is defined by $$\mathrm{p}(x) = ax^3 + 3x^2 + 4ax - 5,$$ where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(\mathrm{p}(x)\).

1. Use the factor theorem to find the value of \(a\). [2]
2. Factorise \(\mathrm{p}(x)\) and hence show that the equation \(\mathrm{p}(x) = 0\) has only one real root. [4]
3. Use logarithms to solve the equation \(\mathrm{p}(6^x) = 0\) correct to 3 significant figures. [2]

Solve the equation \(8^{3-6x} = 4 \times 5^{-2x}\). Give your answer correct to 3 decimal places. [4]

Solve the equation \(\ln(x^3 - 3) = 3 \ln x - \ln 3\). Give your answer correct to 3 significant figures. [3]

Solve the equation \(5^x = 5^{x+2} - 10\). Give your answer correct to 3 decimal places. [3]

The sequence of values given by the iterative formula $$x_{n+1} = \frac{x_n(x_n^2 + 100)}{2(x_n^2 + 25)},$$ with initial value \(x_1 = 3.5\), converges to \(\alpha\).

1. Use this formula to calculate \(\alpha\) correct to 4 decimal places, showing the result of each iteration to 6 decimal places. [3]
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\). [2]

Solve the equation $$2\ln(5 - e^{-2x}) = 1,$$ giving your answer correct to 3 significant figures. [4]

Solve the equation \(\ln(x^2 + 1) = 1 + 2 \ln x\), giving your answer correct to 3 significant figures. [3]

Solve the equation \(2|3^x - 1| = 3^x\), giving your answers correct to 3 significant figures. [4]

Showing all necessary working, solve the equation $$\frac{e^x + e^{-x}}{e^x + 1} = 4,$$ giving your answer correct to 3 decimal places. [5]

Showing all necessary working, solve the equation \(\frac{2e^x + e^{-x}}{e^x - e^{-x}} = 4\), giving your answer correct to 2 decimal places. [4]

Given that \(2^x = \frac{1}{\sqrt{2}}\) and \(2^y = 4\sqrt{2}\),

1. find the exact value of \(x\) and the exact value of \(y\), [3]
2. calculate the exact value of \(2^{y-x}\). [2]

A scientist is using carbon-14 dating to determine the age of some wooden items. The equation for carbon-14 dating an item is given by $$N = k\lambda^t$$ where

• \(N\) grams is the amount of carbon-14 currently present in the item
• \(k\) grams was the initial amount of carbon-14 present in the item
• \(t\) is the number of years since the item was made
• \(\lambda\) is a constant, with \(0 < \lambda < 1\)

1. Sketch the graph of \(N\) against \(t\) for \(k = 1\) [2]

Given that it takes 5700 years for the amount of carbon-14 to reduce to half its initial value,

1. show that the value of the constant \(\lambda\) is 0.999878 to 6 decimal places. [2]

Given that Item A

• is known to have had 15 grams of carbon-14 present initially
• is thought to be 3250 years old

1. calculate, to 3 significant figures, how much carbon-14 the equation predicts is currently in Item A. [2]

Item B is known to have initially had 25 grams of carbon-14 present, but only 18 grams now remain.

1. Use algebra to calculate the age of Item B to the nearest 100 years. [3]

Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which

1. \(3^x = 5\), [3]
2. \(\log_2(2x + 1) - \log_2 x = 2\). [4]

Solve

1. \(5^x = 8\), giving your answer to 3 significant figures, [3]
2. \(\log_2(x + 1) - \log_2 x = \log_2 7\). [3]

A trading company made a profit of £50 000 in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio \(r, r > 1\). The model therefore predicts that in 2007 (Year 2) a profit of £50 000r will be made.

1. Write down an expression for the predicted profit in Year \(n\). [1]

The model predicts that in Year \(n\), the profit made will exceed £200 000.

1. Show that \(n > \frac{\log 4}{\log r} + 1\). [3]

Using the model with \(r = 1.09\),

1. find the year in which the profit made will first exceed £200 000, [2]
2. find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest £10 000. [3]